Shortest distance between 2 skew lines :
`text(Note: )`
`(i)` `2` lines in a plane if not I I must intersect and `2` lines in a plane if not intersecting must be parallel.
Convertely `2` intersecting or parallel lines must be coplanar.
`(ii)` In space, however we come across situation when two lines neither intersect nor II , Two such lines
(like the flight paths of two planes) in space are known as skew lines or non coplanar lines.
`(iii)` `S.D.` between two such skew lines is the segment intercepted betweeen the two lines and perpendicular
to both.
`text(Method 1:) ` Two ways to determine the `S. D.`
`L_1 : vec(r) = vec(a) +lambda vec(p)`
`L_2 : vec(r) = vec(b) + mu vec(q)`
`vec(n) =vec(p) xxvec(q)`
`vec(AB) = (vec(b) - vec(a))`
` SD = | text(Projection of) vec(AB) on vec(n)| = |(vec(AB)* vec(n))/(|vec(n)|) | = | ( ( vec(b) - vec(a)) *(vec(p) xx vec(q)))/( |vec(p)xxvec(q) |) |`
If `S.D. = 0 =>` lines are intersecting and hence coplanar.
`text(Method II :)`
p.v. of `N_1 = vec(a) + lambda vec(p)` ; p.v. of `N_2 = vec(b) + mu vec(q)`
`vec(N_1N_2) = (vec(b) -vec(a)) + (mu vec(q) -lambda vec(p))`
Now `vec(N_1N_2) *vec(p) =0` and `vec(N_1N_2) *vec(q) =0` (two linear equations to get the unique values of `lambda` and `mu` )
One p.v's of `N _1` and `N_2` are known we can also determine the equation to the line of shortest di stance
and the `S.D.`
`(i)` `2` lines in a plane if not I I must intersect and `2` lines in a plane if not intersecting must be parallel.
Convertely `2` intersecting or parallel lines must be coplanar.
`(ii)` In space, however we come across situation when two lines neither intersect nor II , Two such lines
(like the flight paths of two planes) in space are known as skew lines or non coplanar lines.
`(iii)` `S.D.` between two such skew lines is the segment intercepted betweeen the two lines and perpendicular
to both.
`text(Method 1:) ` Two ways to determine the `S. D.`
`L_1 : vec(r) = vec(a) +lambda vec(p)`
`L_2 : vec(r) = vec(b) + mu vec(q)`
`vec(n) =vec(p) xxvec(q)`
`vec(AB) = (vec(b) - vec(a))`
` SD = | text(Projection of) vec(AB) on vec(n)| = |(vec(AB)* vec(n))/(|vec(n)|) | = | ( ( vec(b) - vec(a)) *(vec(p) xx vec(q)))/( |vec(p)xxvec(q) |) |`
If `S.D. = 0 =>` lines are intersecting and hence coplanar.
`text(Method II :)`
p.v. of `N_1 = vec(a) + lambda vec(p)` ; p.v. of `N_2 = vec(b) + mu vec(q)`
`vec(N_1N_2) = (vec(b) -vec(a)) + (mu vec(q) -lambda vec(p))`
Now `vec(N_1N_2) *vec(p) =0` and `vec(N_1N_2) *vec(q) =0` (two linear equations to get the unique values of `lambda` and `mu` )
One p.v's of `N _1` and `N_2` are known we can also determine the equation to the line of shortest di stance
and the `S.D.`
